CS205b/CME306Lecture 61 Springs1.1 1D Mass Spring SystemWe begin with a 1D mass s pring system consisting of a line of n points with mass m and n − 1identical springs connecting them. For a particular spring si, let xi, xi+1, vi, and vi+1be the positionand velocity for the left an d right ends of the spring. Let ∆x = xi+1− xiand ∆v = vi+1− vi. Aswith the simple spring with a fixed endpoint, the force expected will beF = −ks∆xℓ0− 1− kd∆v.Note that translating the spring and examining it at a different point in space do not affect theforce that it applies, so we may observe the sprin g from the position xiof the left en dpoint m ovingat its velocity vi. From this vantage point, the spring looks similar to the simple spr ing w ithits non-fixed endpoint at location ∆x with velocity ∆v. (The system is different, though, in thatneither endpoint is fixed. In particular, it will respon d differently to the force applied by the sprin g.This does not affect the f orce the spring exerts, though.) Note that this is the force the sp ringapplies to the right endpoint. The force applied to the left endpoint is −F as required by Newton’sthird law. The frequency of the s ystem is λ in Hertz (s−1), and the sound speed is c = ℓ0λ, inms−1. The sound speed looks likec = ℓ0sksmℓ0=sksℓ0m.To get more accuracy, we may want to refine this discretization. We would like to double thenumber of points to 2n, which increases the number of springs to 2n − 1 ∼ 2(n − 1). To keep thetotal mass constant, we must replace the n nodes with mass m with 2n nodes with mass ˆm = m/2.To keep the length constant, we must replace the n − 1 springs of length ℓ0with 2n − 1 springs oflengthˆℓ0=n−12n−1ℓ0∼ ℓ0/2. The Young’s modulus ksis a characteristic of the material and doesx1x2x3x4x5s1s2s3s4v1v2v3v4v5Figure 1: Row of springs.1not change. Ignoring the fencepost problem, the sound speed ˆc =pksℓ0/m = c then remains thesame, and the frequency λ doubles toˆλ = 2λ ∼ 2sksmℓ0.Note that halving the s pring length but keeping the s ame sound speed forces the frequency todouble, since inf ormation must travel through twice as many springs in the same amount of time.Since ∆tλ ≈ 1, the stable time step size ∆t is halved toˆ∆t = ∆t/2 due to the refinement. Thefraction x/ℓ0that the material compresses or stretches does not change under the refinement, sinceboth x and ℓ0are both halved. The resulting elastic force and acceleration for a refined spring areˆF = −ksx/2ℓ0/2− 1= F ˆa =Fm/2= 2a.In particular, the force is unchanged, but the acceleration of the ind ividual particles doubles. Finallykd0is a property of the material and should not change, so thatˆkd= kd0sksm/2ℓ0/2= kd.In summary, if the resolution of the discretization is doubled, the various parameters of the systemchange as follows:ˆm =m2ˆℓ0=ℓ02ˆks= ksˆkd= kdˆc = cˆλ = 2λˆ∆t =∆t2ˆF = F ˆa = 2a1.2 Zero Length SpringSimply setting ℓ0= 0 in the simple spring model causes p roblems. However, when modelingmaterials, this value is reached through the limit n → ∞, ℓ0→ 0, and m → 0.Another potential situation where a zero-length spring might be used is to connect two objects.One solution to this problem is to place the connection points an arbitrary distance ℓ0/2 insideeach object. Letting s be the amount the joint is separated, the distance between the ends of thespring is x = ℓ0+ s, and its derivative is where v = x′= s′. The spr ing force isF = −ksxℓ0− 1− kdv = −ksℓ0+ sℓ0− 1− kdv = −ksℓ0s − kdv.The resulting system is then the same as before, except that it lacks the inhomogeneous term, andis sv!′= 0 1−ksmℓ0−kdm! sv!.We can now choose ksand ℓ0, or we could jus t factor out the arbitrary parameter ℓ0into the s pringco efficient withˆks=ksℓ0F = −ˆksx − kdv kd= kd0pmks.2(a) Edge springs do not preventshear.(b) Edge and shear springs do notprevent bending.(c) Bending springs between twotriangles.Figure 2: Mass spring model for cloth.1.3 2D Mass Spring System in 3DCloth may be modeled as a 2D surface that lives in 3D. The cloth surface might be discretized asa Cartesian grid of masses and springs, with masses at the corners connected along the edges bysprings. The spr ings effectively restrict stretching and compression along the axes.This cloth discretization lacks forces that are able to combat shear in the cloth. If a piece ofcloth were modeled as a single square, one could fold the cloth into a line that is tw ice the edgelength. Shearing the entire Cartesian grid of cloth has the same effect, illustrated in Figure 2(a).One way of solving this problem is to add springs along the diagonals of the s quares in the clothgrid. One might choose to place springs on all of the left diagonals, all of the right diagonals, somemixture of left and right diagonals, or both left and right diagonals. Shear forces are different fromstretching forces. Cloth tends to be very resistant to stretch but shears relatively easily, so shearsprings are typically much weaker than edge springs. An alternative approach would be to usefinite elements for the cloth, but for now we shall stick to springs.This cloth model still has a major problem. This panel of cloth has no resistance to bendingalong the lines of springs that run along the axial directions, as in Figure 2(b). This may befixed with additional springs that correct bending, such as the vers ion obtained here by connectingopposing vertices of adjacent triangles (the shear springs create triangles). Bending forces in clothare also typically very weak.These bend ing springs have a couple potential limitations. If the desired angle between twotriangles is nonzero, the bending spring will be unable to apply a force that would tend to correctthe bend if the two triangles are coplanar or are bent in the wrong direction. Another issue withthis model is that bending forces become very weak when the triangles are nearly coplanar. Thesetwo issues can be alleviated by connecting the shared edge between the triangles to the bendingspring with yet another spring. This configuration is show n in Figure 2(c).This setup works well in practice but may fail to apply forces in the right direction for someconfigurations of the cloth triangles. Note that the two triangles (formed by edge and shear springs)along with the original bending spring form a